These remarkable well-posedness results for Bayesian inverse problems for PDEs rely on the well-posedness of the underlying forward problem, often requiring that the mapping \(\mathcal {L}\)  in (REF ) is Lipschitz continuous in suitable metrics and converting this Lipschitz continuity into stability results for the posterior measure with respect to perturbations in the measurements, see [1]} for a survey of these results and their applications to a variety of PDEs. More recently in [2]}, [3]}, these Lipschitz continuity assumptions on the forward map \(\mathcal {L}\)  in (REF ), have been considerably relaxed. In particular, under suitable assumptions on the measurement noise \(\eta \)  in (REF ), mere existence and measurability of the forward map suffices for the well-posedness of the underlying Bayesian inverse problem [2]}.
